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 2005-06-29, 17:22 #1 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22×33×19 Posts Prime free sequence. Much more is known about how far apart primes are than about how close they are. By choosing the number n as large as you want how can we have a prime free sequence of consecutive whole numbers as long as you want? Mally
2005-06-29, 19:16   #2
maxal

Feb 2005

111111002 Posts

Quote:
 Originally Posted by mfgoode Much more is known about how far apart primes are than about how close they are. By choosing the number n as large as you want how can we have a prime free sequence of consecutive whole numbers as long as you want? Mally
Take n!+2, n!+3, n!+4, ..., n!+n for any integer n > 1.
They all are composite since n!+i has non-trivial divisor i.

 2005-06-29, 21:38 #3 Numbers     Jun 2005 Near Beetlegeuse 38810 Posts As this is so obviously true, and n + 1 + 2 + 3... is an infinite series, doesn't that imply that there is out there somewhere an infinite gap with no primes in it? I find it quite difficult to reconcile that idea with the proof that the primes themselves are infinite. So where does this infinite gap fit in?
2005-06-29, 22:19   #4
jinydu

Dec 2003
Hopefully Near M48

2·3·293 Posts

Quote:
 Originally Posted by Numbers As this is so obviously true, and n + 1 + 2 + 3... is an infinite series, doesn't that imply that there is out there somewhere an infinite gap with no primes in it? I find it quite difficult to reconcile that idea with the proof that the primes themselves are infinite. So where does this infinite gap fit in?
In order for that to be true, n (in fact, I think you mean n!) would have to be infinite, which would make it not an integer. No such infinite gap exists.

Last fiddled with by jinydu on 2005-06-29 at 22:20

2005-06-29, 22:22   #5
R.D. Silverman

Nov 2003

746010 Posts

Quote:
 Originally Posted by Numbers As this is so obviously true, and n + 1 + 2 + 3... is an infinite series, doesn't that imply that there is out there somewhere an infinite gap with no primes in it? I find it quite difficult to reconcile that idea with the proof that the primes themselves are infinite. So where does this infinite gap fit in?
Gibberish. Illucid.

(0) From where did you get the expression n + 1 + 2 + 3 +....??? It has
ZERO connect with any prior discussion.

(1) There is no such thing as an infinite prime. There are infinitely *many*,
but all primes are *finite*

(2) There is no such thing as an "infinite gap". The gap (equal to the
difference) between any two integers is also an integer. All integers are
finite. The gap between primes can be arbitrarily large. That is, for any
integer M, you can find a gap between primes that is larger than M. Period.

 2005-06-29, 22:31 #6 Ken_g6     Jan 2005 Caught in a sieve 5×79 Posts Welcome to the wonderful (and crazy) world of infinity and limits! The increase of n! is much faster than the increase of the gap size, so there's always plenty of room outside the gaps left for primes. Here's a proof that there are infinitely many primes, which actually uses these gaps!
 2005-06-29, 22:40 #7 robert44444uk     Jun 2003 Oxford, UK 19·103 Posts Arbitrary large Take an integer = x#, where # is the symbol primorial, such that x#= 2*3*5*7*...*x x can be any prime number, and there are an infinite number of those. Lets take a really big x (i.e. largest possible prime ie. infinitely large) The gap between x# and x#+x+2 is prime free, and this gap is arbitrarily and infinitely large. Regards Robert Smith "Play with fire, its safer than playing with infinity"
2005-06-29, 22:56   #8
JHansen

Apr 2004
Copenhagen, Denmark

11610 Posts

Quote:
 Originally Posted by robert44444uk x can be any prime number, and there are an infinite number of those. Lets take a really big x (i.e. largest possible prime ie. infinitely large) The gap between x# and x#+x+2 is prime free, and this gap is arbitrarily and infinitely large.
Now you are just trying to tease Dr. Silverman. That is not nice of you.

--
Cheers,
Jes

2005-06-29, 23:40   #10
ewmayer
2ω=0

Sep 2002
República de California

2×3×29×67 Posts

Quote:
 Originally Posted by robert44444uk Take an integer = x#, where # is the symbol primorial, such that x#= 2*3*5*7*...*x x can be any prime number, and there are an infinite number of those. Lets take a really big x (i.e. largest possible prime ie. infinitely large)
No - it makes no sense to speak of primes (or composites, or even integers, for that matter) being "infinitely large". Primes, composites and integers are all *numbers* - infinity is not a number, although by convention it can be manipulated in some ways like finite numbers can.

Quote:
 The gap between x# and x#+x+2 is prime free, and this gap is arbitrarily and infinitely large.
Again, you're blithely mixing concepts that sound superficially similar but are profoundly different. "Arbitrarily" in your sense means "you give me any finite natural N, I can find a prime-free gap whose length exceeds N." But any such gap will clearly not be infinite in length, since the very definition of a gap between primes implies the existence of a next-larger prime, i.e. one which bounds the gap from above. And even without an explicit gap we know that gaps cannot be arbitrarily large with respect to the primes bracketing them, since by Bertrand's Postulate (a.k.a. Chebyshev's theorem) if n > 1, then there is always at least one prime p such that n < p < 2*n. So one first needs to be precise about what one means by "arbitrarily". And irrespective of whether one is referring to the absolute or relative size of prime gaps, "arbitrarily" in this context does not mean "infinitely."

2005-06-30, 11:02   #11
tom11784

Aug 2003
Upstate NY, USA

2×163 Posts

Quote:
 Originally Posted by robert44444uk The gap between x# and x#+x+2 is prime free...
not to be picky, but only x#+2 to x#+x+1 need be composite (take x=5)
this is either a gap of x terms, or a subsequence of a larger gap

Last fiddled with by tom11784 on 2005-06-30 at 11:03

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